This is a phase I pharmacology study of intravenous BCX-34 (Peldesin) in up to twelve patients with refractory T-cell and non-T-cell malignancies. Three thirty-minute intravenous infusions separated by at least four days with intra-patient dose increase in the first three evaluated patients will be given; in the absence of dose-limiting toxicity, the second three evaluable patients will receive two increasing dose levels over thirty minutes and one four-hour infusion at a lower dose. The objective is to determine if, at or below the MTD (and within the fluid volume limitations of drug administration), there is a progressive increase in plasma deoxyguanosine with increasing drug dose and if the maximum effect is related primarily to the peak plasma level or AUC of BCX-34. Secondarily, the effect of increasing dose on plasma inosine and urine inosine excretion will be assessed. Multiple blood and urine samples will be obtained for drug levels, inosine, and deoxyguanosine in association with each infusion. Anti-tumor activity will be noted.